Theorem fvprif 13376

Description: The value of the pair function at an element of 2o. (Contributed by Mario Carneiro, 14-Aug-2015.)

Assertion

Ref	Expression
fvprif	|- ( ( A e. V /\ B e. W /\ C e. 2o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` C ) = if ( C = (/) , A , B ) )

Proof of Theorem fvprif

Step	Hyp	Ref	Expression
1		fvpr0o 13374	. . . . 5 |- ( A e. V -> ( { <. (/) , A >. , <. 1o , B >. } ` (/) ) = A )
2	1	3ad2ant1 1042	. . . 4 |- ( ( A e. V /\ B e. W /\ C e. 2o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` (/) ) = A )
3	2	adantr 276	. . 3 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = (/) ) -> ( { <. (/) , A >. , <. 1o , B >. } ` (/) ) = A )
4		simpr 110	. . . 4 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = (/) ) -> C = (/) )
5	4	fveq2d 5631	. . 3 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = (/) ) -> ( { <. (/) , A >. , <. 1o , B >. } ` C ) = ( { <. (/) , A >. , <. 1o , B >. } ` (/) ) )
6	4	iftrued 3609	. . 3 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = (/) ) -> if ( C = (/) , A , B ) = A )
7	3, 5, 6	3eqtr4d 2272	. 2 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = (/) ) -> ( { <. (/) , A >. , <. 1o , B >. } ` C ) = if ( C = (/) , A , B ) )
8		fvpr1o 13375	. . . . 5 |- ( B e. W -> ( { <. (/) , A >. , <. 1o , B >. } ` 1o ) = B )
9	8	3ad2ant2 1043	. . . 4 |- ( ( A e. V /\ B e. W /\ C e. 2o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` 1o ) = B )
10	9	adantr 276	. . 3 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = 1o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` 1o ) = B )
11		simpr 110	. . . 4 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = 1o ) -> C = 1o )
12	11	fveq2d 5631	. . 3 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = 1o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` C ) = ( { <. (/) , A >. , <. 1o , B >. } ` 1o ) )
13		1n0 6578	. . . . . 6 |- 1o =/= (/)
14	13	neii 2402	. . . . 5 |- -. 1o = (/)
15	11	eqeq1d 2238	. . . . 5 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = 1o ) -> ( C = (/) <-> 1o = (/) ) )
16	14, 15	mtbiri 679	. . . 4 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = 1o ) -> -. C = (/) )
17	16	iffalsed 3612	. . 3 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = 1o ) -> if ( C = (/) , A , B ) = B )
18	10, 12, 17	3eqtr4d 2272	. 2 |- ( ( ( A e. V /\ B e. W /\ C e. 2o ) /\ C = 1o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` C ) = if ( C = (/) , A , B ) )
19		elpri 3689	. . . 4 |- ( C e. { (/) , 1o } -> ( C = (/) \/ C = 1o ) )
20		df2o3 6576	. . . 4 |- 2o = { (/) , 1o }
21	19, 20	eleq2s 2324	. . 3 |- ( C e. 2o -> ( C = (/) \/ C = 1o ) )
22	21	3ad2ant3 1044	. 2 |- ( ( A e. V /\ B e. W /\ C e. 2o ) -> ( C = (/) \/ C = 1o ) )
23	7, 18, 22	mpjaodan 803	1 |- ( ( A e. V /\ B e. W /\ C e. 2o ) -> ( { <. (/) , A >. , <. 1o , B >. } ` C ) = if ( C = (/) , A , B ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 713   /\ w3a 1002   = wceq 1395   e. wcel 2200   (/)c0 3491   ifcif 3602   {cpr 3667   <.cop 3669   ` cfv 5318   1oc1o 6555   2oc2o 6556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-id 4384  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-res 4731  df-iota 5278  df-fun 5320  df-fv 5326  df-1o 6562  df-2o 6563

This theorem is referenced by: (None)